Second-order rotatable design

Box, G.E.P., and Behnken, D.W. (1960a), Simplex-Sum Designs A Class of Second Order Rotatable Designs Derivable from those of First Order, Ann. Math. Statist., 31, 838-864. 

Second-order Rotatable Design (Box-Wilson Design)... 

Previous research has indicated that for mathematical modeling of the process it is necessary to use second-order rotatable design. The design matrix with experimental outcomes are given in Table 2.153. 

Reference [56] states that these are the smallest second-order rotatable designs, with a smaller number of trials than central composite rotatable designs. Of special use are designs that are made by vertices of hexagons with central points n0>l Fig. 2.58. 

Another alternative to the 3 full factorial is the Box-Behnken design (Box and Behnken [19]). These designs are a class of incomplete three-level factorial designs that either meet, or approximately meet, the criterion of rotatability. A Box-Behnken design for p=3 variables is shown in Table 2.7. This design will estimate the ten coefficients of the second-order... 

Central composite rotatable designs Quantitative Regression models of second order... 

Hartley s design with only 27 trials should first of all be used for k=5. Box s rotatable design also deserves attention. A comparison of rotatable designs of second order with D-optimal and other designs shows that a rotatable design may be applied where limits of an experimental region are given by a sphere, i.e. in cases when a researcher is only interested in the response surface in the vicinity of the... 

Rotatable designs are most efficient for k=3. Rotatable designs of second order are not orthogonal and they do not minimize the variance of estimates of regression coefficients. They are efficient in solving research problems when trying to find an optimum. 

To optimize the process of isomerization of sulphanylamide from Problem 2.6, a screening experiment has been done by the random balance method. Factors X1 X2 and X3 have been selected for this experiment. Optimization of the process is done by the given three factors at fixed values of other factors. To obtain the second-order model, a central composite rotatable design has been set up. Factor-variation levels are shown in Table 2.148. The design of the experiment and the outcomes of design points are in Table 2.149. 

This example refers to response dependence on two factors (k=2). Orthogonal second-order design in this case, according to Table 2.164, has nine design points (N=9). The design matrix with outcomes of design points is shown in Table 2.165. The same case has been elaborated in the previous section, in Example 2.43, by application of rotatable second-order design. However, the connection between coded and real values of factors for the same null point is now different ... 

Equation (2.171) obtains different forms depending on a specific experiment or design of experiments and number of replications. To replicate all trials of a design evenly (even number of replications) and for N0>1, we use Eq. (2.171). In the case of a rotatable second-order design, when trials are replicated in all points the same number of times, Eq. (2.171) becomes ... 

Equation (2.173) is used for rotatable designs of second and third order when trials are replicated only in null point. In the case of a full factorial experiment or regular fractional replicas, we use ... 

Rotatable second-order design Only in design center (2.168) and (2.130)... 

Rotatable second-order design Equal replication (2.167) and (2.128) N, E (yu- u)... 

An experiment with composite rotatable second-order design, where trials have been replicated only in design center, is given in Example 2.45. Five factors and system response (N=32 k=5 n0=6 Nxn=N-(n0-l)=27) have been studied in the experiment. Outcomes are given in Table 2.140. By processing experimental outcomes, we have obtained the regression model (2.97). To check lack of fit of the obtained regression model, we used data from Table 2.1. 

To reach a second-order model by a mathematical theory of experiment, designs of second order experiments may be applied, as described in sects. 2.3.2, 2.3.3 and 2.3.4. Noncomposite designs such as simplex sum rotatable designs (SSRD) pentagonal or hexagonal types (k=2) with central points are analyzed in this case (Figs. 2.57 and 2.58.). 

---